A description of the equidistribution properties of Hecke operators of p-adic elliptic curves. The most difficult case, of supersingular elliptic curves, is analyzed using Lubin-Katz theory of the canonical subgroups, and the Gross-Hopkins period map coming from Serre-Tate’s theory of deformations of Abelian varieties. A key ingredient is a version of Linnik’s equidistribution theorem for certain p-adic quaternion algebras that is of independent interest. This is a joint work with Sebastian Herrero and Ricardo Menares